Editing Function field (scheme theory) (section)

== General case ==

The trouble starts when ''X'' is no longer integral. Then it is possible to have [[zero divisor]]s in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the [[total quotient ring]], that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.

The correct solution is to proceed as follows:

:For each open set ''U'', let ''S<sub>U</sub>'' be the set of all elements in Γ(''U'', ''O<sub>X</sub>'') that are not zero divisors in any stalk ''O<sub>X,x</sub>''. Let ''K<sub>X</sub><sup>pre</sup>'' be the presheaf whose sections on ''U'' are [[Localization of a ring|localizations]] ''S<sub>U</sub><sup>−1</sup>''Γ(''U'', ''O<sub>X</sub>'') and whose restriction maps are induced from the restriction maps of ''O<sub>X</sub>'' by the universal property of localization. Then ''K<sub>X</sub>'' is the sheaf associated to the presheaf ''K<sub>X</sub><sup>pre</sup>''.